Lecture 03

Bill Perry

Lecture 2: Review

  • We covered:

    • data wrangling and types of variable names

    • meta data

    • project design

    • summary statistics

    • graphing the mean and standard error graphs

    • pipes and %>% or |> and how to group_by

Our last graph

Lecture 3: How to deal with data wrangling

Introduction to probability distributions

  • What is a frequency distribution?
  • What is a probability distribution?
  • Distributions for variables and for statistics

Estimation

  • Populations and samples
  • Parameters and statistics

we are going to use some sculpin data that is real!

Wikipedia

Lecture 3: Frequency distributions

Example data we will use will be a combination of data from Toolik Alaska LTER

source

We will specifically look at fishes like

Lake Trout

source

Grayling

Slimy Sculpin

source

Lecture 3: Frequency distributions

  • Data - has been cleaned in terms of lake names and species names
  • Slimy Sculpin - Toolik Lake
# A tibble: 1 × 4
   mean    sd    se count
  <dbl> <dbl> <dbl> <int>
1  51.7  12.0 0.834   208

Lecture 3: Frequency Distributions

What is a frequency distribution?

  • Display of number of observations in certain intervals
  • e.g., the number of sculpin per interval in Toolik Lake
  • as a table like below or histogram
# A tibble: 28 × 2
   length_bin     n
   <fct>      <int>
 1 [11,13]        4
 2 (19,21]        1
 3 (23,25]        1
 4 (27,29]        2
 5 (29,31]        2
 6 (31,33]        1
 7 (33,35]        4
 8 (35,37]        3
 9 (37,39]        7
10 (39,41]        9
# ℹ 18 more rows

Lecture 3: Frequency Distributions

The alternative is to use a histogram

  • the y axis is the count
  • the x axis is the bin range
  • each bin 0 - 5 and 5 - 10 and 10 - 15 or as you choose
  • in ggplot the code looks like
dataframe %>% ggplot(aes(thing_to_count))+
  geom_histogram( 
    binwidth = increments_to_work_with
    )

Lecture 3: Frequency Distributions

What happens as sample size changes…

  • Sampls size
    • Low sample number - 15
    • High sample number - 70
  • Frequency distribution takes on “bell-shape”…

Lecture 3: Probability distributions

Can we make assumption about distribution of random variable weight in population?

Probability distribution:

  • theoretical frequency distribution in population

Lecture 3: Probability distributions

  • For continuous random var: probability density function (PDF)
  • PDF: mathematical expression of probabilities associated with getting certain values of random variable
  • Area under curve = 1
  • i.e., probability of length between 10 and 80 = 1

Lecture 3: Probability distributions

Now we could look at a lot of different ranges of lengths

  • probability of the length larger than the mean

  • probability of the length larger than 70 mm

  • probability of the length between two numbers

Lecture 3: Probability distributions

  • Usually need to know probability distribution of random variables in statistical analyses
  • Can define many distributions; some do reasonable job especially whit continuous variables
  • Different distributions for continuous, discrete variables like a single die

Lecture 3: Probability distributions

Normal (Gaussian): symmetrical, bell-shaped

  • Defined in terms of mean and variance (μ, 𝜎2)

  • SND (z-distribution) has mean μ=0 , 𝜎2 =1

\(f(y) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(y - \mu)^2}{2\sigma^2}}\)

Lecture 3: Probability distributions

  • Lognormal: right-skewed distribution
  • Logarithm of random variable is normally distributed
  • Common in biology.
  • Why would this occur or be common in biology?

Lecture 3: Probability distributions

Binomial (multinomial):

  • probability of event that have two outcomes (heads/ tails, dead/alive)
  • Defined in terms of “successes” out of set number of trials

In large number of trials: approximately normal distribution

Lecture 3: Probability distributions

Poisson: occurrences of (rare) event in time/space

  • E.g., number of
    • Taraxacum officinale - common dandelion in quadrat
    • copepod eaten per minute
    • cells in field of view
  • Measures Probability(y= certain integer value)

    • defined in terms of μ or mean

    • Right-skewed at small μ

    • more symmetrical at higher μ

Lecture 3: Data gathering - managing

Also have distributions of test statistics

Test statistics:

  • summary values calculated from data used to test hypotheses
  • is your result due to chance?

Different test statistics:

  • different, well-defined distributions
  • allows estimation of probabilities associated with results
  • Examples:
    • z-distribution, student’s t-distribution, χ2-distribution, F-distribution

Lecture 3: Samples and populations

Inferential statistics:

  • inference from samples to populations

Statistical population:

  • All possible observations of interest
  • Normally: populations too large to census

Populations are defined in time + space

Examples of statistical populations from you research area?

Lecture 3: Samples and populations

Key characteristic of sample is

  • size (n observations; n = sample size)

Characteristics of population - called parameters

  • Parameters - Greek letters

Characteristics of samples - statistical estimates of parameters

  • statistics= Latin letters

Random sampling crucial for

sample -> population

inference statistics -> parameters

Lecture 3: Parameters and statistics

Two main kinds of summary statistics: - center and spread

Center: - Mean (µ, ȳ): sum of sampled values divided by n - Mode: the most common number in dataset - Median: middle measurement of data; = mean for normal distributions

Mean

\(\mu = \frac{\sum\limits_{i=1}^{n} Y_i}{n}\)

Formula for n odd

\(\text{median = } Y\_{(n+1)/2}\)

Formula for n even

\(\text{median = }\frac{Y_{n/2} + Y_{(n/2)+1}}{ 2}\)

Lecture 3: Parameters and statistics

Spread

  • Range: from highest and lowest observation
  • Variance (σ2, s2): sum of squared differences of observations from mean, divided by n-1

E.g., fish lengths = 20, 30, 35, 24, 36 g

# A tibble: 1 × 1
   mean
  <dbl>
1    29

\(s^2 = \sum_{i=1}^{n} \frac{(y_i - \bar{y})^2}{n-1}\)

Lecture 3: Parameters and statistics

Spread

(20 -29)^2+ (30 -29)^2 + (35 -29)^2 + (24 -29)^2 + (36 -29)^2 = 57,104

192 / (5-1) = 48 mm^2 Problem: weird units!

# A tibble: 1 × 2
   mean variance
  <dbl>    <dbl>
1    29       48

Lecture 3: Parameters and statistics

Spread

  • Standard Deviation(σ, s): square root of variance.

    • In same units as observations

    • In example: √48 = 6.9 mm

  • Coefficient of variation: SD as % of mean.
    • Useful for comparing spread in samples with different means
    • In example: (6.9/29)*100= 23.8 %

\(s^2 = \sqrt{\sum_{i=1}^{n} \frac{(y_i - \bar{y})^2}{n-1}}\)

\(\text{Coefficient of variation} = \frac{S}{\bar{Y}} \times 100\)

Lecture 3: Estimation

Problem: - don’t know the values of parameters

Goal: - estimate parameters from empirical data (samples)

3 general methods of parameter estimation: - Maximum Likelihood Estimation (MLE) - Ordinary Least Squares (OLS) - Resampling techniques

  • MLE general method to estimate parameters in a way that maximizes the likelihood of the observed data given the parameter values.

  • aims to find the parameter values that make the observed data most probable under the assumed statistical model.

  • OLS specific method to estimate parameters of a linear regression model.

  • minimizes the sum of the squared differences between observed and predicted values

Lecture 3: Estimation